Measure functional differential equations and functional dynamic equations on time scales.

*(English)*Zbl 1239.34076Measure functional differential equations of the form
\[
x(t)=x(t_0)+\int_{t_{0}}^tf(x_s,s)dg(s), ~~ t\in [t_0,t_0+\sigma],
\]
are studied, where the integral on the right-hand side is the Kurzweil-Stieltjes integral with respect to a nondecreasing function \(g.\)

The relation between measure functional differential equations and generalized ordinary differential equations is described. It is explained also that impulsive functional differential equations represent a special case of measure functional differential equations. Moreover the relation between functional dynamic equations on time scales and measure functional differential equations is established.

Using the existing theory of generalized ordinary differential equations, results on the existence and uniqueness of a solution and on the continuous dependence of a solution on parameters for both, measure functional differential equations and functional dynamic equations on time scales, are obtained.

The relation between measure functional differential equations and generalized ordinary differential equations is described. It is explained also that impulsive functional differential equations represent a special case of measure functional differential equations. Moreover the relation between functional dynamic equations on time scales and measure functional differential equations is established.

Using the existing theory of generalized ordinary differential equations, results on the existence and uniqueness of a solution and on the continuous dependence of a solution on parameters for both, measure functional differential equations and functional dynamic equations on time scales, are obtained.

Reviewer: Sotiris K. Ntouyas (Ioannina)

##### MSC:

34K05 | General theory of functional-differential equations |

34K45 | Functional-differential equations with impulses |

34N05 | Dynamic equations on time scales or measure chains |

34K33 | Averaging for functional-differential equations |

##### Keywords:

measure functional differential equations; functional dynamic equations; generalized ordinary differential equations; existence and uniqueness; continuous dependence on a parameter; periodic averaging
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\textit{M. Federson} et al., J. Differ. Equations 252, No. 6, 3816--3847 (2012; Zbl 1239.34076)

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